If x is measured in degree then`(d)/(dx)` (cos x ) is equal to

Options

A. `- sinx`
B. `-(180)/pi sin x`
C. `-(pi)/(180) sin x`
D. sin x

Answer

To find the derivative of \( \cos x \) where \( x \) is measured in degrees, we can follow these steps:

### Step 1: Convert degrees to radians
Since the derivative of trigonometric functions is typically calculated in radians, we need to convert \( x \) from degrees to radians. The conversion formula is:
\[ \text{radians} = \frac{\pi}{180} \times \text{degrees} \]
Thus, if \( x \) is in degrees, we can express it in radians as:
\[ x_{\text{radians}} = \frac{\pi x}{180} \]

### Step 2: Write the cosine function in radians
Now we can express \( \cos x \) in terms of radians:
\[ \cos x = \cos\left(\frac{\pi x}{180}\right) \]

### Step 3: Differentiate the cosine function
To differentiate \( \cos\left(\frac{\pi x}{180}\right) \), we apply the chain rule. The derivative of \( \cos u \) is \( -\sin u \) where \( u = \frac{\pi x}{180} \). Therefore, we have:
\[ \frac{d}{dx} \cos\left(\frac{\pi x}{180}\right) = -\sin\left(\frac{\pi x}{180}\right) \cdot \frac{d}{dx}\left(\frac{\pi x}{180}\right) \]

### Step 4: Differentiate the inner function
Now we differentiate the inner function \( \frac{\pi x}{180} \):
\[ \frac{d}{dx}\left(\frac{\pi x}{180}\right) = \frac{\pi}{180} \]

### Step 5: Combine the results
Putting it all together, we get:
\[ \frac{d}{dx} \cos x = -\sin\left(\frac{\pi x}{180}\right) \cdot \frac{\pi}{180} \]
Thus, the final result is:
\[ \frac{d}{dx} \cos x = -\frac{\pi}{180} \sin\left(\frac{\pi x}{180}\right) \]

### Final Answer
\[ \frac{d}{dx} \cos x = -\frac{\pi}{180} \sin\left(\frac{\pi x}{180}\right) \]

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